Three-Dimensional Bodies of Minimum Total Drag in Hypersonic Flow

The problem of constructing three-dimensional bodies of minimum total drag is studied within the framework of a local interaction model. Under certain assumptions, this model can be adopted to describe the distributions of both pressure and skin friction on the body during its high-speed motion through gases and dense media. Without any constraints on the possible drag law within the scope of the accepted model, the optimum shapes providing the minimum drag are found without any simplifying assumptions regarding their geometry. It is shown that, for a given base area and specified limitations on the body size, one can construct an infinite number of optimum forebody shapes. It is proved that the desired shapes are formed by combinations of surface parts whose normal makes a certain constant angle with the direction of motion. The optimum angle is determined by the velocity and medium characteristics in terms of the constants of the drag law. A method of optimum shape design is proposed; in particular, it allows one to construct optimum bodies like missiles with aft feather and optimum bodies with a circular base. All the bodies constructed have the same minimal total drag for the given base area. Even for asymmetrical bodies, the acting force has no component in a plane perpendicular to the direction of motion. Special attention is paid to the particular case of the minimum drag body design in hypersonic flow, when the pressure on the body is specified by the Newton formula. A comparative study of the results obtained for Newtonian flow shows that the proposed shapes are more effective in providing a drag reduction than bodies found to be optimum in earlier studies under special simplifying assumptions.     

车身非光滑表面凹坑排布对气动性能的影响

受仿生学非光滑旋成体减阻启发,以SAE(美国机动车工程师协会)标准模型为研究对象,采用CFD(计算流体力学)数值模拟方法,在SAE模型顶部布置不同排布形式和不同排布密度的凹坑单元,研究其对车身气动性能的影响.通过比较各模型的尾流、气流速度、压力场、湍流动能等流场性能指标,分析非光滑表面减阻机理以及造成各模型流场性质差异的原因.计算结果显示:当凹坑型非光滑单元以矩形排布时模型具有最小的气动阻力,且气动阻力随着凹坑密度的增加而减小,减阻率最高达到4.1%.     

An elementary solution of a variational problem of aerodynamics

The problem of Miele of finding the transversal contour of a conical body of given length and base area so as to minimize the total drag in hypersonic flow is solved here without making use of the usual tools of the calculus of variations. The solution makes use of the isoperimetric inequality, together with the usual elementary methods for determining the minimum of a function of one real variable.     

解任意形状扁轴对称体Stokes流动的连续奇点分布法

本文以Sampson球形无穷级数作为基本奇点,采用分段等强度和分段二次抛物分布两种体内连续分布法解任意形状扁轴对称体的Stokes流动.通过扁球的无界绕流问题,对这两种方法的收敛性,精度和适用范围做了检验和比较.结果表明,在一定的范围内,无论是阻力系数或压力分布,它们的计算结果都和精确解符合得很好,而且,随着分布函数逼近程度的提高,其收敛性得到改善,适用范围也随之扩大.作为一般算例,分别用这两种方法解决了卡西尼扁卵形体的绕流问题,得到了一致的结果.最后,用分段二次连续分布法计算了具有一定生理意义的红细胞体的Stokes流动,首次得到了它的阻力系数和表面压力分布.     

Überschallknall und Widerstand eines vorne spitzen Rotationskörpers in einer schweregeschichteten Atmosphäre

Summary J. L. Ryhming [5] concluded that the minimum-boom body for given bow-shock wave drag and maximum body thickness is also the minimum drag body due to the bow shock for a given maximum body thickness. Here it is shown that the above conclusion is only valid for asymptotic distances. The geometry of the minimum-boom body depends on the distance for which one wants to minimize the sonic boom.     

Stokes drag on axially symmetric bodies: a new approach

In this paper a new approach to evaluate the drag force in a simple way on a restricted axially symmetric body placed in a uniform stream (i) parallel to its axis, (ii) transverse to its axis, is advanced when the flow is governed by the Stokes equations. The method exploits the well-known integral for evaluating the drag on a sphere. The method not only provides the value of the drag on prolate and oblate spheroids and a deformed sphere in axial flow which already exists in literature but also new results for a cycloidal body, an egg shaped body and a deformed sphere in transverse flow. The salient results are exhibited graphically. The limitations imposed on the analysis because of the lack of fore and aft symmetry in the case of an eggshaped body is also indicated. It is also seen that the analysis can be extended to calculate the couple on a body rotating about its axis of symmetry.     

One nonlinear variational problem of the theory of cavitating profiles

In this paper we study the limiting values of the lift and drag coefficients of profiles in the Helmholtz-Kirchhoff (infinite cavity) flow. The coefficients are based on the wetted arc length of profile surfaces. Namely, for a given value of the lift coefficient we find minimum and maximum values of the drag coefficient. Thereby we determine maximum and minimum values of the lift-to-drag ratios.     

Optimal body design using localized interaction models

The problem of the designing bodies of minimum drag for a specified area of the base and a specified area of the (“windward”) surface around which the flow occurs is considered in the approximation of an arbitrary localized interaction model. New necessary conditions for minimum drag are obtained which are stronger than the Legendre condition. It is shown that, in the approximation used, the optimal configurations in the general case contain end faces and cylindrical segments of the boundary extremum, which appear due to the existence of limits of applicability of local models. It is established that the solutions previously obtained are incomplete. A complete solution of the problem is constructed.     

On the “equivalence” rule for flows of perfect gas : PMM vol. 38, n 6, 1974, pp. 1004–1014

An extension of the equivalence of “area” rule [1, 2] is presented. The rule was initially derived for stationary flows of perfect (inviscid and non-heat-conducting) gas past slender fine pointed bodies (or blunted bodies in the hypersonic flow case) whose transverse dimensions are small in comparison with their length. According to that rule the wave drag of a three-dimensional body is equal to the wave drag of an axisymmetric body with the same distribution of cross-sectional areas along the axis. The rule is extended here to stationary and nonstationary flows past nonslender bodies and to internal flows, using the procedure of averaging with respect to the angular variable of a cylindrical system of coordinates. That procedure is, strictly speaking, valid for nearly axisymmetric bodies. However the numerical solutions obtained by the authors for a fairly wide range of external and internal problems show that the generalized equivalence rule is applicable to substantially nonaxisymmetric configurations (^*) (see next page).     

The motion of a submerged sphere in a liquid under an ice sheet

The solution of the linear steady problem of the flow of an inviscid, incompressible and infinitely deep liquid around a sphere under an ice sheet, which is modelled by a thin elastic stressed plate of constant thickness is constructed. Special cases of this problem are the motion of a submerged sphere under broken ice, a membrane, and also under the free surface both in the presence and absence of capillary effects. The method of multipole expansions is used in the framework of the linear potential wave theory. The hydrodynamic loads (the wave drag and the buoyancy) acting on the body and also the distribution of the deflections of the ice sheet are calculated as a function of the body velocity, the ice thickness and the value of the compressing or stretching forces. It is shown that all the flow characteristics depend considerably on the ratio of the body velocity and the critical velocity of flexural-gravitational waves.     

The conical deformation of a delta wing with sonic leading edges

The optimal conical deformation of a delta wing with sonic leading edges is determined in a linear formulation of the problem. The drag of the wing caused by the creation of the lifting force is taken as the objective function. It is established that a superelliptical distribution of the local angle of attack over the wing span corresponds to the minimum drag. A representation in the form of a hypergeometric function is found for the directrix of the wing in a cross section. The results obtained are compared with the results of a numerical investigation within the Euler model.     

A study of an arbitrary Stokes flow past a fluid coated sphere in a fluid of a different viscosity

A general method to discuss the problem of an arbitrary Stokes flow (both axisymmetric and non-axisymmetric flows) of a viscous, incompressible fluid past a sphere with a thin coating of a fluid of a different viscosity is considered. We derive the expressions for the drag and torque experienced by the fluid coated sphere and also discuss the conditions for the reduction of the drag on the fluid coated sphere. In fact, we show that the drag reduces compared to the drag on a rigid sphere of the same radius when the unperturbed velocity is either harmonic or purely biharmonic, i.e., of the form ${r^2\vec{\textbf{v}}}$ , where ${\vec{\textbf{v}}}$ is a harmonic function. Previously Johnson (J Fluid Mech 110:217–238, 1981), who considered a uniform flow showed that the drag on the fluid coated sphere reduces compared to the drag on the uncoated sphere when the ratio of the surrounding fluid viscosity to the fluid-film viscosity is greater than 4. We show that this result is true when the undisturbed velocity is harmonic or purely biharmonic, uniform flow being a special case of the former. However, we illustrate by an example that the drag may increase in a general Stokes flow even if this ratio is greater than 4. Moreover, when the unperturbed velocity is harmonic or purely biharmonic, and the ratio of the surrounding fluid viscosity to the fluid-film viscosity is greater than 4 for a fixed value of the viscosity of the ambient fluid, we determine the thickness of the coating for which the drag is minimum.     

Potential flow past a porous body with a core of different permeability

It is well known that a uniform flow past a non-permeable rigid body does not exert a total force upon the surface of the body, however this is not the case when the body is permeable. Power et. al. (1984, 1986) first solved the problem of uniform potential flow past a two-dimensional permeable circular cylinder, with constant permeability, and found that the exterior flow exerts a drag force upon the surface of the cylinder independent of its size and secondly the problem when the uniform potential flow past a porous sphere, with constant permeability, in this case the exterior flow exerts a drag force on the sphere which is linearly dependent on the radius of the sphere. Here we will present the solution of two problems, a uniform potential flow past a porous circular cylinder and past a porous sphere, for each case the porous body is composed of two materials with different permeabilities. In both cases the total force exerted by the exterior flow upon the body is dependent on the thickness of the porous materials, and in the limit when the two permeabilities are equal, the previous results, circular cylinder and sphere, with constant permeability, are recovered. Atlhough, the mathematics involved in the solution of the present problem is simple, due to the nice boundary geometry of the bodies, the final expression for the total force found in each case is quite interesting on the way it depends on the permeability relation, in particular, in the limiting cases of a porous body with solid or hollow core.     

Mathematical modeling of turbulent fiber suspension and successive iteration solution in the channel flow

The modified Reynolds mean motion equation of turbulent fiber suspension and the equation of probability distribution function for mean fiber orientation are firstly derived. A new successive iteration method is developed to calculate the mean orientation distribution of fiber, and the mean and fluctuation-correlated quantities of suspension in a turbulent channel flow. The derived equations and successive iteration method are verified by comparing the computational results with the experimental ones. The obtained results show that the flow rate of the fiber suspension is large under the same pressure drop in comparison with the rate of Newtonian fluid in the absence of fiber suspension. Fibers play a significant role in the drag reduction. The amount of drag reduction augments with increasing of the fiber mass concentration. The relative turbulent intensity and the Reynolds stress in the fiber suspension are smaller than those in the Newtonian flow, which illustrates that the fibers have an effect on suppressing the turbulence. The amount of suppression is also directly proportional to the fiber mass concentration.     

On the minimization of the product of the powers of several integrals

This paper considers the minimization of the product of the powers ofn integrals, each of which depends on a functiony(x) and its derivative . The necessary conditions for the extremum are derived within the frame of the Mayer-Bolza formulation of the calculus of variations, and it is shown that the extremal arc is governed by a second-order differential equation involvingn undetermined multipliers related to the unknown values of the integrals. After the general solution is combined with the definitions of the multipliers and the end conditions, a system ofn+2 algebraic equations is obtained; it involvesn+2 unknowns, that is, then undetermined multipliers and two integration constants.The procedure discussed here can be employed in the study of shapes which are aerodynamically optimum at supersonic, hypersonic, and free-molecular flow velocities, that is, wings and fuselages having the maximum lift-to-drag ratio or the minimum drag. The problem of a slender body of revolution having the minimum pressure drag in Newtonian hypersonic flow is developed as an example. First, a general solution is derived for any pair of conditions imposed on the length, the thickness, the wetted area, and the volume. Then, a particular case is treated, that in which the thickness and the wetted area are given, while the length and the volume are free; the shape minimizing the pressure drag is a cone.This research, supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-828-67, is a condensed version of the investigation described in Ref. 1. The author is indebted to Messrs. H. Y. Huang, J. C. Heideman, and J. N. Damoulakis for analytical and numerical assistance.     

Hypersonic shapes of maximum drag for a given heat-transfer rate

The problem of determining the shape of a blunt, axisymmetric body which maximizes the drag for a given heat-transfer rate and diameter is considered. For blunt bodies, pressure drag predominates and is estimated from modified Newtonian flow considerations. With regard to heat transfer, it is assumed that the body is operating in the range of hypersonic speeds where the radiative heating rate can be neglected with respect to the convective heating rate. The latter is estimated from the boundary-layer analysis due to Lees. Optimum power-law shapes as well as variational shapes are determined and are shown to yield almost identical results. For low-to-moderate values of the convective heat-transfer parameter, the optimum shape is very flat and is approximately a one-half power-law shape; in this range, spherical segments are approximately one-half power-law shapes and, hence, are nearly maximum drag shapes. There exists a maximum value of the convective heat-transfer parameter for which maximum drag shapes exist, and the corresponding optimum shape is a cone, or a power-law shape of exponent unity. This limiting shape is shown to be that which maximizes the convective heat-transfer rate for a given diameter.This research was supported in part by NASA-Manned Spacecraft Center under Contract No. NAS-6963. The authors are indebted to Dr. John J. Bertin for helpful discussions and suggestions concerning heattransfer aspects of this paper.     

An invariant variational problem of the laminar boundary layer

A variational problem on minimizing, by normal injection into a laminar boundary layer, the Newtonian drag of a blunt cylindrical body in a supersonic flow of an ideal gas is considered, taking into account the limitation on the power of the system to control the injection. Using the first integral obtained, the order of the conjugate system is reduced, which enables an effective algorithm to be constructed for finding the optimal control using the grid method. The results of a computational experiment are presented, according to which the gains in the values of the drag functional for the optimal controls obtained reach 65% compared with a uniform injection law.     

一个新的边界层方程及其在形状控制中的应用

本文给出固壁边界上(即一个二维流形上) 的流体速度梯度和压力的二阶偏微分方程, 从而也给出边界上法向应力, 以及流体中运动物体所受的阻力和升力的计算公式. 本方法的创新在于边界上法向速度梯度不是通过在边界层内速度梯度的数值微分达到, 而是通过它与其他变量一起作为一组偏微分方程的解而得到, 证明边界层方程组的适定性问题, 并且给出解关于边界形状的Gâteaux 导数所满足的偏微分方程. 本文将本方法应用于飞机外形的形状最优控制, 给出阻力泛函关于形状第一变分的可计算形式. 数值例子表明, 用本方法得到的阻力精度比通用程序得到要高.     

The design of symmetric, optimal supersonic and hypersonic flow profiles for arbitrary isoperimetric conditions

A simple and accurate approach to the design of symmetric profiles which are optimal in the supersonic and hypersonic flow with an attached shock is developed. Besides dimensional constraints, the bodies being optimized can satisfy arbitrary isoperimetric conditions. The approach which has been developed uses a modification of the “shock-expansion” method (SEM). The modified shock-expansion method (MSEM), unlike SEM, does not lead to a physically absurd result, that is, to a finite change in the flow parameters when the slope of the contour is solely changed at the leading point of the body. This makes MSEM suitable for solving two-dimensional variational problems in gas dynamics, by reducing any of them to a certain extension of the Lagrange problem for systems which are described by ordinary differential equations. The possibilities of the approach which has been developed are illustrated using examples of profiles which achieve a minimum wave drag coefficient, C_x. Profiles designed using the MSEM are compared with those obtained using the Newtonian model and linear theory and with wedges while the C_x values found for them using the above-mentioned approximate models and MSEM are compared with the results of the numerical integration of Euler's equations.     

The legendre condition in optimum problems of supersonic gasdynamics : PMM vol. 39, n 6, 1975, pp. 1032–1042

The necessary Legendre condition for problems of optimum (in the sense of minimum wave drag) supersonic flow past bodies is obtained. Plane and axisymmetric flows are considered on the assumption of imposition of isoperimetric constraints of a general form. Shock-free flows and flows with attached shock waves are investigated. The method here proposed is used for deriving the second order condition in the particular case when it is possible to pass to the reference contour, and which has been earlier obtained by Shmyglevskii [1] and then by Guderley and others [2].